Applications of patching the coherent cohomology of modular curves

TL;DR

This paper applies advanced patching techniques to the coherent cohomology of modular curves, establishing new algebraic properties and density results for crystalline deformation rings and modular forms.

Contribution

It introduces novel applications of the Taylor--Wiles--Kisin patching method to modular curves, proving Cohen--Macaulayness and multiplicity-one results in new contexts.

Findings

Proves a multiplicity-one result for patched modules using the q-expansion principle.

Shows a partial normalization of the crystalline deformation ring is Cohen--Macaulay.

Establishes Zariski density of crystalline points in characteristic p.

Abstract

In this paper, we apply the Taylor--Wiles--Kisin patching method to the coherent cohomology of modular curves at minimal level. We establish a multiplicity-one result for the patched module by the $q$-expansion principle and show that a certain partial normalization of the crystalline deformation ring is Cohen--Macaulay. As applications, we prove new cases where crystalline deformation rings are Cohen--Macaulay, establish a Zariski density result for crystalline points in characteristic $p$, and prove a multiplicity-one result for Serre's mod-$p$ quaternionic modular forms.